Pauli exclusion principle

This question might be beyond our current knowledge but i want to make sure. The reason why fermions do not overlap is due to the pauli exclusion principle which states that one fermion cannot occupy the same state as another fermion. a quark is 10-18m and a proton is 10-15m, 3 orders of magnitude. since mt everest is 103, it follows that a quark is to a proton, what a human is to mt everest sizewise. there are just 3 quarks jostling around in the proton, to say nothing of all the empty space inside the quark (in fact if string theory is right, there are 15 orders of magnitude inside the quark of empty space, but we'll ignore that for now, since it's not proven) so here's my question: how does one fermion exclude the occupancy of another fermion if the space inside the proton is so empty. it's as if we have a sphere or radius 1000 and in it 3 quarks of radius 1 represent the entire sphere. i'm pretty sure the answer to this is not currently known, but i want to make sure.

category justification: i put this in the nuclear physics since it deals with quarks.

You may have heard that a proton is made from three quarks. Indeed here are several pages that say so. This is a lie — a white lie, but a big one. In fact there are zillions of gluons, antiquarks, and quarks in a proton. The standard shorthand, “the proton is made from two up quarks and one down quark”, is really a statement that the proton has two more up quarks than up antiquarks, and one more down quark than down antiquarks. To make the glib shorthand correct you need to add the phrase “plus zillions of gluons and zillions of quark-antiquark pairs.” Without this phrase, one’s view of the proton is so simplistic that it is not possible to understand the LHC at all.